4.5 Point null hypotheses for the normal distribution

4.5.1 Calculation of the Bayes’ factor

Suppose  is a vector of independently  random variables, and that is known. Because of the remarks at the end of the last section, we can work entirely in terms of the sufficient statistic

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We have to make some assumption about the density  of θ under the alternative hypothesis, and clearly one of the most natural things to do is to suppose that this density is normal, say  . Strictly, this should be regarded as a density on values of θ other than  , but when probabilities are found by integration of this density, the odd point will make no difference. It will usually seem sensible to take  as, presumably, ...

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